Weakly 3-Connected Graphs

Abstract

A 3-connected graph $G$ is weakly 3-connected if, for every edge $e$ of $G$, at most one of $G\backslash e$ and $G/e$ is 3-connected. The main result of this paper is that any weakly 3-connected graph can be reduced to $K_4$ by a sequence of simple operations. This extends a result of Dawes [5] on minimally 3-connected graphs.